Vector Calculus

Vector calculus is the fundamental language of mathematical physics. It pro­ vides a way to describe physical quantities in three-dimensional space and the way in which these quantities vary. Many topics in the physical sciences can be analysed mathematically using the techniques of vector calculus. These top­ ics include fluid dynamics, solid mechanics and electromagnetism, all of which involve a description of vector and scalar quantities in three dimensions. This book assumes no previous knowledge of vectors. However, it is assumed that the reader has a knowledge of basic calculus, including differentiation, integration and partial differentiation. Some knowledge of linear algebra is also required, particularly the concepts of matrices and determinants. The book is designed to be self-contained, so that it is suitable for a pro­ gramme of individual study. Each of the eight chapters introduces a new topic, and to facilitate understanding of the material, frequent reference is made to physical applications. The physical nature of the subject is clarified with over sixty diagrams, which provide an important aid to the comprehension of the new concepts. Following the introduction of each new topic, worked examples are provided. It is essential that these are studied carefully, so that a full un­ derstanding is developed before moving ahead. Like much of mathematics, each section of the book is built on the foundations laid in the earlier sections and chapters.

Contenu

1. Vector Algebra.- 1.1 Vectors and scalars.- 1.1.1 Definition of a vector and a scalar.- 1.1.2 Addition of vectors.- 1.1.3 Components of a vector.- 1.2 Dot product.- 1.2.1 Applications of the dot product.- 1.3 Cross product.- 1.3.1 Applications of the cross product.- 1.4 Scalar triple product.- 1.5 Vector triple product.- 1.6 Scalar fields and vector fields.- 2. Line, Surface and Volume Integrals.- 2.1 Applications and methods of integration.- 2.1.1 Examples of the use of integration.- 2.1.2 Integration by substitution.- 2.1.3 Integration by parts.- 2.2 Line integrals.- 2.2.1 Introductory example: work done against a force.- 2.2.2 Evaluation of line integrals.- 2.2.3 Conservative vector fields.- 2.2.4 Other forms of line integrals.- 2.3 Surface integrals.- 2.3.1 Introductory example: flow through a pipe.- 2.3.2 Evaluation of surface integrals.- 2.3.3 Other forms of surface integrals.- 2.4 Volume integrals.- 2.4.1 Introductory example: mass of an object with variable density.- 2.4.2 Evaluation of volume integrals.- 3. Gradient, Divergence and Curl.- 3.1 Partial differentiation and Taylor series.- 3.1.1 Partial differentiation.- 3.1.2 Taylor series in more than one variable.- 3.2 Gradient of a scalar field.- 3.2.1 Gradients, conservative fields and potentials.- 3.2.2 Physical applications of the gradient.- 3.3 Divergence of a vector field.- 3.3.1 Physical interpretation of divergence.- 3.3.2 Laplacian of a scalar field.- 3.4 Curl of a vector field.- 3.4.1 Physical interpretation of curl.- 3.4.2 Relation between curl and rotation.- 3.4.3 Curl and conservative vector fields.- 4. Suffix Notation and its Applications.- 4.1 Introduction to suffix notation.- 4.2 The Kronecker delta ?ij.- 4.3 The alternating tensor ?ijk.- 4.4 Relation between ?ijk and ?ij.- 4.5 Grad, div and curl in suffix notation.- 4.6 Combinations of grad, div and curl.- 4.7 Grad, div and curl applied to products of functions.- 5. Integral Theorems.- 5.1 Divergence theorem.- 5.1.1 Conservation of mass for a fluid.- 5.1.2 Applications of the divergence theorem.- 5.1.3 Related theorems linking surface and volume integrals.- 5.2 Stokes's theorem.- 5.2.1 Applications of Stokes's theorem.- 5.2.2 Related theorems linking line and surface integrals.- 6. Curvilinear Coordinates.- 6.1 Orthogonal curvilinear coordinates.- 6.2 Grad, div and curl in orthogonal curvilinear coordinate systems.- 6.2.1 Gradient.- 6.2.2 Divergence.- 6.2.3 Curl.- 6.3 Cylindrical polar coordinates.- 6.4 Spherical polar coordinates.- 7. Cartesian Tensors.- 7.1 Coordinate transformations.- 7.2 Vectors and scalars.- 7.3 Tensors.- 7.3.1 The quotient rule.- 7.3.2 Symmetric and anti-symmetric tensors.- 7.3.3 Isotropic tensors.- 7.4 Physical examples of tensors.- 7.4.1 Ohm's law.- 7.4.2 The inertia tensor.- 8. Applications of Vector Calculus.- 8.1 Heat transfer.- 8.2 Electromagnetism.- 8.2.1 Electrostatics.- 8.2.2 Electromagnetic waves in a vacuum.- 8.3 Continuum mechanics and the stress tensor.- 8.4 Solid mechanics.- 8.5 Fluid mechanics.- 8.5.1 Equation of motion for a fluid.- 8.5.2 The vorticity equation.- 8.5.3 Bernoulli's equation.- Solutions.